This equation (REF ) belongs to the family of completely integrable equations and, for a broad class of initial conditions, its integrability is realised via the inverse scattering transform (IST) method [1]}.
The inverse scattering theory associates a soliton of the KdV equation with a point of discrete spectrum \(\lambda =\lambda _n\) , of the Schrödinger operator
\(\mathcal {L}= -\partial _{xx}^2 - u(x,t) \, .\) 
